Trees and Fractals

This picture was taken from my backyard on a grey December morning. I love the dramatic look of bare branches silhouetted against the sky. Like many other things in nature, the shapes of trees exhibit striking mathematical patterns. In fact, the verb "branch" describes the mathematical process that produces the shapes.

I notice that branches are formed by a step-by-step process of splitting into smaller and smaller parts.I notice that I can create "stick drawings" that look a lot like trees by using simple branching patterns.I notice that the branching process seems like a mathematical pattern.I wonder how many branchings usually occur before you get to a leaf (or the place where a leaf would be).I wonder if the mathematical details of branching patterns are different for different types of trees.I notice that the branchings of the tree seem to show a mixture of predictability and randomness.I wonder what other things in nature show repeated branching patterns like this.I notice that math is everywhere–even in places where I never expect to find it.

By repeating simple mathematical rules on smaller and smaller scales, you can create symmetrical designs that look remarkably like real trees. If you insert a little randomness into the process, they become even more realistic looking! The types of shapes that emerge from repeating patterns across a range of scales are known as fractals. You can see many examples of fractal images of trees (natural and human-made) on Google!

Notice how the examples tend to look much the same when you "zoom in" or "zoom out" on them. This is called self-similarity. Can you invent your own processes for creating self-similar tree fractals? Can you do it starting with shapes other than line segments?

By the way, did the power line in the picture catch your attention? Teaching a computer to recognize something like this requires some relatively sophisticated mathematics—yet our brains do it naturally. It is as if we are doing complex mathematical calculations in our brains all the time without realizing it!